<h2>Problem 261</h2>
<div style="color:#666;font-size:80%;">23 October 2009</div><br />
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<p>Let us call a positive integer <var>k</var> a <var>square-pivot</var>, if there is a pair of integers <var>m</var> <img src='images/symbol_gt.gif' width='10' height='10' alt='&gt;' border='0' style='vertical-align:middle;' /> 0 and <var>n</var> <img src='images/symbol_ge.gif' width='10' height='12' alt='&ge;' border='0' style='vertical-align:middle;' /> <var>k</var>, such that the sum of the (<var>m</var>+1) consecutive squares up to <var>k</var> equals the sum of the <var>m</var> consecutive squares from (<var>n</var>+1) on:</p>
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(<var>k</var>-<var>m</var>)<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + ... + <var>k</var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = (<var>n</var>+1)<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + ... + (<var>n</var>+<var>m</var>)<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />.</div>

<p>Some small square-pivots are
<ul><li><b>4</b>: 3<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + <b>4</b><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />
 = 5<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /></li>
<li><b>21</b>: 20<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + <b>21</b><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 29<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /></li>
<li><b>24</b>: 21<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 22<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 23<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + <b>24</b><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 25<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 26<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 27<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /></li>
<li><b>110</b>: 108<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 109<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + <b>110</b><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 133<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + 134<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /></li></ul></p>

<p>Find the sum of all <b>distinct</b> square-pivots <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 10<img src="" style="display:none;" alt="^(" /><sup>10</sup><img src="" style="display:none;" alt=")" />.</p>
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